The first step in deriving information-theoretic lower bounds for minimax risk is to reduce the above estimation problem to a testing problem [1]}, [2]}, [3]}. To do so, we need to construct an index set \({\mathcal {V}}\)  with \(|{\mathcal {V}}|<\infty \)  and a family of distributions \(\lbrace P_v, v\in {\mathcal {V}}\rbrace \subseteq {\mathcal {P}}\)  such that \(\ell (\theta (P_v), \theta (P_{v^{\prime }}))\ge 2\tau \)  for all \(v \ne v^{\prime }\)  in \({\mathcal {V}}\)  for some \(\tau >0\) . The canonical testing problem is then defined as follows: Nature chooses a random variable \(V\)  uniformly at random from \({\mathcal {V}}\) , and then conditioned on \(V= v\) , the samples \(X^n\)  are drawn i.i.d. from \(P_v\) , denoted by \(X^n\sim P^{\otimes n}_v\) . Each \(X_i\)  is then fed to a mechanism \({\mathsf {K}}_i\)  to generate \(Z_i\) . It is well-known [3]}, [2]}, [1]} that \(\mathcal {R}_n({\mathcal {P}}, \ell )\ge \tau \mathsf {P}_{\mathsf {e}}(V|X^n)\) , where \(\mathsf {P}_{\mathsf {e}}(V|X^n)\)  denotes the probability of error in guessing \(V\)  given \(X^n\) . Replacing \(X^n\)  by its \((\varepsilon , \delta )\) -privatized samples \(Z^n\)  in this result, one can obtain a lower bound on \(R_n({\mathcal {P}}, \ell , \varepsilon , \delta )\)  in terms of \(\mathsf {P}_{\mathsf {e}}(V|Z^n)\) . Hence, the remaining challenge is to lower-bound \(\mathsf {P}_{\mathsf {e}}(V|Z^n)\)  over the choice of mechanisms \(\lbrace {\mathsf {K}}_i\rbrace \) .
There are numerous techniques for this objective depending on \({\mathcal {V}}\) . We focus on two such approaches, namely Le Cam's and Fano's method, that bound \(\mathsf {P}_{\mathsf {e}}(V|Z^n)\)  in terms of total variation distance and mutual information and hence allow us to invoke Lemmas REF  and REF .
